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Hah! It depends on the width of the implied 4th! When you carefully consider what the tests are for the P12, the P15 and the implied P4 you should realize that if the P4 is pure, the P12 and the P15 can also be pure. The algebraic difference of the beatrates of the P12 and the P15 equal the beatrate of the implied P4. This is only true when the common note is on top.

Let me add this. One more little known things about the effects of iH on beatrates is that the beatrate of P4 does not double each octave. It can take a couple of octaves to double. So the beat difference between P12s and P15s does not increase much toward the treble.

Originally Posted By: Mark Davis

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And if one is tuning a pure 12th tuning in the treble and high treble, does this not increase the stretch more than say, a just 4:1 octave throughout the treble and high treble? or does this vary from piano to piano?

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Kees explained it well:

Originally Posted By: DoelKees

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Unrelated to my writeup, note that the default tuning curve in Tunelab has pure 6:3 octaves in the bass, and pure 4:1 double octaves in the treble, with a smooth transition in-between. In particular the 4:2 always becomes narrow in the higher octaves (around C6). A pure 12th tuning results in narrow 4:2 and wide 4:1.

Kees

Originally Posted By: Mark Davis

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What do your octaves and double octaves sound like when tuning this way?

As you go up the treble, a beat can be detected, but is not obvious unless the temperment is not strictly equal. It can be looked at as a side benefit. Since you are closer to the edge of acceptable stretch, a note that is a little off an octave below the P12 that is being tuned will not sound right when listening to the octave.

I used to tune the treble without as much stretch and have smoother sounding octaves and 15ths, but the top octaves just didn't sound like they were at the right pitch when music that uses a great deal of the keyboard is played. I tried stretching by tuning wide octaves but my results were sometimes, ahem, "haphazard." Again, no complaints from customers, but I did not like the inconsistencies. So like the story of Goldilocks and the Three Bears, pure octaves was not enough, stretched octaves was too much, but pure 12ths was just right.

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Jeff DeutschlePart-Time TunerWho taught the first chicken how to peck?

@Jeff : " It depends on the width of the implied 4th! When you carefully consider what the tests are for the P12, the P15 and the implied P4 you should realize that if the P4 is pure, the P12 and the P15 can also be pure. The algebraic difference of the beatrates of the P12 and the P15 equal the beatrate of the implied P4. "

something wrong in your reasoning, to me, or works only in absence of iH,

Seem to me that the definition of "pure" is what bothers me.a 4:2 octave to me, imply yet some stretch (as a 2:1 in the end) . Cordier (pure 5th tuning ) agreed in the end that the good term is "acoustically pure" as this is a slightly different concept, implying more than the simple 3:1 4:1 or 2:1 relation.

Edited by Kamin (01/11/1301:05 PM)

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Professional of the profession.

I wish to add some kind and sensitive phrase but nothing comes to mind.!

There are probably others scratching their heads on this one, too. It is a relationship that Mark C. works with.

Consider these three notes: A3, D4 and A5. There are three intervals possible: the A3-D4 P4, the D4-A5 P12 and the A3-A5 P15.

Consider a normal situation where A3-D4 P4 is 1bps wide. This means the fourth partial of A3 is at 880hz and the third partial of D4 is at 881hz. (These are approximate non-iH theoretical pitches but it does not matter. 1 bps wide is 1 bps wide regardless of the actual hz involved.) When A5 is tuned, if it is at 880hz the result is a pure P15 and a narrow P12, if it is at 881hz the result is a wide P15 and a pure P12, and if it is at 880.5hz the result is a wide P15 and a narrow P12 that beat at the same speed. Regardless of where A5 is tuned, the algebraic difference of the beatrates (+ for wide, - for narrow) will equal the beatrate of the implied P4. They could even both be wide or narrow: If A5 = 886hz then (+6bps) – (+ 5bps) = + 1bps.

Now consider an abnormal situation where A3-D4 P4 is pure. Both the fourth partial of A3 and the third partial of D4 are at 880hz. If A5 is tuned to 880hz, both the P12 and the P15 will be pure.

Using F3 as the standard test note for these intervals may make it easier to understand… or not!

This is why I say that difference between the P12 and P15 depends on the implied P4.

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Jeff DeutschlePart-Time TunerWho taught the first chicken how to peck?

The advantage we have when tuning is that the high iH slow the beats of 4ths . that is what allows the octave to be enlarged without 4th being too noisy.

So "acoustically" a stretched 4th may sound pure, if you see what I mean, and a tempered 5th or 12th as well

When tuning , I consider all intervals as simple additions or substractions, so I may be reasoning on a no IH based logic, I agree, but this is based to what is perceived, for instance, beats are evaluated without locating them at a particular level, as if I was using pure frequencies.

Edited by Kamin (01/11/1303:01 PM)

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Professional of the profession.

I wish to add some kind and sensitive phrase but nothing comes to mind.!

Jeff, if you stick to pure 12th you are providing a fundation for the high treble, the tuning is in fact straight and sound just in the two modes, arpeggios and octaves.

Just for fun try to locate the exact pitch of the notes in the 5-6th octave region just based on the resonant spot obtained from the 12th, and the double.

May be you will notice that your note have a raise in color at some point.

When tested you possibly will find an equilibrium between the 15th and the double octave (same beat speed)

I will not promise it works anytime but it works.

I am advantaged with so called "perfect pitch" that allow me to place the note more or less well once I have heard the others (and also with the slow motion technique, hence the habit to master progressive slow pitch changes at a a slow speed)

the minor chord inversion (M3+4th) played with the double octave is supposed to stay quiet enough, active but not noisy nor acid.

May be the resonant spot lower the perception of the speed of the M3 and the 6th played together...

When the tuning is good, that chord is showing the wanted pitch of the double octave as a laser light .

I also like the fact that the tuning is lively.

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Professional of the profession.

I wish to add some kind and sensitive phrase but nothing comes to mind.!

Are we all on the same page? Are we all speaking of tuning from completed unisons or are some speaking of tuning a piano that has a temperament strip in tha whole piano, thus tuning and refering to and from only single strings?

Let me try to make something clear. If a 12th is tuned pure, the 15th will beat the same speed as the implied 4th. And when a 15th is tuned pure the 12th will beat the same speed as the implied 4th. Of course this only pertains to when the common note is on top. It cannot be any other way. Inharmonicity and temperment strips have nothing to do with it.

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Jeff DeutschlePart-Time TunerWho taught the first chicken how to peck?